Section: New Results

Algebraic Technique For Estimation, Differentiation And Its Applications

Algebraic technique is the other tool we develop for providing finite-time convergence.

• The integer order differentiation by integration method based on the Jacobi orthogonal polynomials for noisy signals was originally introduced by Mboup, Join and Fliess. The paper [35] proposes an extention of this method from the integer order to the fractional order to estimate the fractional order derivatives of noisy signals. Two fractional order differentiators are deduced from the Jacobi orthogonal polynomial filter, using the Riemann-Liouville and the Caputo fractional order derivative definitions respectively. Exact and simple formulas for these differentiators are given by integral expressions. Some error bounds are provided for the corresponding estimation errors. The noise error contribution due to a large class of stochastic processes is studied in discrete case.

• Armed with structures, group sparsity can be exploited to improve the performance of adaptive estimation. In the paper [45] , the adaptive estimation algorithm for cluster structured sparse signals, called A-CluSS, is proposed. In particular, a hierarchical Bayesian model is built, where both sparse prior and cluster structured prior are exploited simultaneously. The adaptive updating formulas for statistical variables are obtained via the variational Bayesian inference and the resulted algorithms can adaptively estimate the cluster structured sparse signals without knowledge of block size, block numbers and block locations. In [75] , a group sparse regularized least-mean-square (LMS) algorithm is proposed to cope with the identification problems for multiple/multi-channel systems. An iterative online algorithm is proposed via proximal splitting method.